Equivalent higher order linear and non-linear systems

The object of this study is to put forth the concept of equivalence of classes of linear and non-linear systems of higher order. In particular, the equivalence of classes of non-linear (non-autonomous/autonomous) systems of order n described by partial/ordinary differential equations with corresponding classes of linear systems of order (n + 1) is established through a differential transformation of the dependent variable. In view of the fact that the resulting linear systems are amenable to existing state-space techniques, this approach can be expected to be of value in the study of many non-linear problems arising in a variety of disciplines. The possible applications of the technique are illustrated with an example.     

Equivalent linear models for non-linear non-autonomous systems

Equivalence between a class of non-linear non-autonomous systems of second order and a linear model of lower order is established through a differential transformation relation. It is shown that this equivalence can be established only under a certain constraint on the non-linear functional parameters of the given system. The equivalence automatically leads to the first integral which then can be analyzed further to obtain the response of the system. The feasibility of obtaining closed form solutions through such analysis is illustrated by considering certain sub-classes of systems. Further, the practical value of the technique is demonstrated through an example.     

Spectral analysis of non-linear systems involving square-law operations

This paper contains a number of useful theoretical formulas to analyze the frequency domain properties of Gaussian input data passing through non-linear square-law systems. Special bispectral density functions are defined and applied that are functions of a single variable. From measurements of input data and output data only, results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non-linear square-law systems. Non-linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non-linear operations. Together with ordinary coherence functions, a measured output spectrum for these models can be decomposed into three components representing the linear operations, the non-linear operations, and the remaining uncorrelated noise effects. This material indicates also how to analyze other types of non-linear models by employing similar techniques.     

Equilibrium point damage prognosis models for structural health monitoring

Most structural dynamic systems are of high order; however, they often exhibit phenomena that can be dealt with effectively using low order models. This paper presents a method for describing certain kinds of damage evolution in mechanical systems. The method relies on a simple principle that as damage evolves in a structural dynamic system, the damage indicator (i.e., diagnostic feature) behaves like a stable quasi-stationary equilibrium point in a subsidiary non-linear bifurcating system within the so-called damage center manifold. It is shown that just as linear normal modes govern the behavior of linear structures with idealized damping, so too do non-linear normal forms govern the evolution of damage within structures in many instances. The method is justified with citations from the literature on certain types of mechanical failure and then applied in an experimental case involving reversible damage in a bolted fastener. Off-line experiments on a rotorcraft fuselage show that the evolution of damage is sensitive to both temporal and spatial bifurcation parameters. A diagnostic sensing strategy whereby output-only transmissibility features are used to decrease the order of high order structural dynamic measurements is also described.     

Experimental and Numerical Study of Structural Identification Using Non-Linear Resonant Decay Method

One of the practical approaches in identifying structures is the non-linear resonant decay method which identifies a non-linear dynamic system utilizing a model based on linear modal space containing the underlying linear system and a small number of extra terms that exhibit the non-linear effects. In this paper, the method is illustrated in a simulated system and an experimental structure. The main objective of the non-linear resonant decay method is to identify the non-linear dynamic systems based on the use of a multi-shaker excitation using appropriated excitation which is obtained from the force appropriation approach. The experimental application of the method is indicated to provide suitable estimates of modal parameters for the identification of non-linear models of structures.     

Linear and non-linear energy barriers in systems of interacting single-domain ferromagnetic particles

A comparative analysis between linear and non-linear energy barriers used for modeling statistical thermally-excited ferromagnetic systems is presented. The linear energy barrier is obtained by new symmetry considerations about the anisotropy energy and the link with the non-linear energy barrier is also presented. For a relevant analysis we compare the effects of linear and non-linear energy barriers implemented in two different models: Preisach-Néel and Ising-Metropolis. The differences between energy barriers which are reflected in different coercive field dependence of the temperature are also presented.     

Analysis of the faster-than-Nyquist optimal linear multicarrier system

Faster-than-Nyquist signalization enables a better spectral efficiency at the expense of an increased computational complexity. Regarding multicarrier communications, previous work mainly relied on the study of non-linear systems exploiting coding and/or equalization techniques, with no particular optimization of the linear part of the system. In this article, we analyze the performance of the optimal linear multicarrier system when used together with non-linear receiving structures (iterative decoding and direct feedback equalization), or in a standalone fashion. We also investigate the limits of the normality assumption of the interference, used for implementing such non-linear systems. The use of this optimal linear system leads to a closed-form expression of the bit-error probability that can be used to predict the performance and help the design of coded systems. Our work also highlights the great performance/complexity trade-off offered by decision feedback equalization in a faster-than-Nyquist context.     

Weyl quantization and star products

Many non-linear classical mechanical systems arise as the symplectic reductions of linear systems. The star products on the corresponding quantized algebras can be derived from the Weyl-Moyal product on the algebras of the linear systems. An algebraic approach to Berezin quantization is sketched.     

On the occurrence of quasiperiodic motion on three tori

Two systems of three coupled non-linear oscillators are considered, one with linear and one with non-linear coupling. It is shown, with the help of numerical calculations of a number of indicators, that, in both cases, quasiperiodic flow can typically occur on three tori and survive under small changes in the parameters of the two systems. This result could be of consequence in our understanding of non-linear phenomena including turbulence.     

Reprint of : Quantum interference in a Cooper pair splitter: The three sites model

New generation of Cooper pair splitters defined on hybrid nanostructures are devices with high tunable coupling parameters. Transport measurements through these devices revealed clear signatures of interference effects and motivated us to introduce a new model, called the 3-sites model. These devices provide an ideal playground to tune the Cooper pair splitting (CPS) efficiency on demand, and displays a rich variety of physical phenomena. In the present work we analyze theoretically the conductance of the 3-sites model in the linear and non-linear regimes and characterize the most representative features that arise by the interplay of the different model parameters. In the linear regime we find that the local processes typically exhibit Fano-shape resonances, while the CPS contribution exhibits Lorentzian-shapes. Remarkably, we find that under certain conditions, the transport is blocked by the presence of a dark state. In the non-linear regime we established a hierarchy of the model parameters to obtain the conditions for optimal efficiency.     

FREQUENCY DOMAIN ARX MODEL AND MULTI-HARMONIC FRF ESTIMATORS FOR NON-LINEAR DYNAMIC SYSTEMS

Non-linear dynamic systems respond at frequencies other than the excitation frequency; however, standard frequency response function estimators for linear systems do not accommodate this harmonic distortion. A new multi-harmonic frequency response function estimator that utilizes discrete frequency models for non-linear systems is introduced here. The multi-harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain. Overdetermined, least-mean-squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators. The resulting multi-harmonic frequency response function models are non-parametric (e.g., vary with amplitude) when linear functions are used and parametric when non-linear functions are used. A new sensitive indicator for experimentally characterizing non-linearity is introduced.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

A system of strings and springs in three dimensions

A three-dimensional array of classical strings and springs oscillating in three dimensions is investigated with combined analytical-numerical tools. The strings are continuous and linear. The long time behavior of the system changes dramatically when the springs are changed from linear to non-linear. The equipartition of energy, even among the particles, is badly violated. The power spectrum is consistent with chaotic motion. The maximal Lyapunov exponent is vanishing within our approximation. The most remarkable finding is a certain pairing of time averages for the particles, which occurs for many initial conditions in the non-linear case.     

Differential transform method for solving linear and non-linear systems of partial differential equations

In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

EFFECTS OF BASE POINTS AND NORMALIZATION SCHEMES ON THE NON-LINEAR NORMAL MODES OF CONSERVATIVE SYSTEMS

In this paper, two factors that affect the behaviors of the non-linear normal modes (NNMs) of conservative vibratory systems are investigated. The first factor is the base points (which are equivalent to Taylor series expanding points) of the non-linear normal modes and the second one is the normalization schemes of the corresponding linear modes. For non-linear systems, in general only the approximated NNM manifolds are obtainable in practice, so different base points may lead to different forms of NNM oscillators and different normalization schemes lead to different forward and backward transformations which in turn affect the numerical computation errors. Three different kinds of base points and two different normalization schemes are adopted for comparison respectively. Two examples of non-linear systems with two and three degrees of freedom, respectively, are given as illustration. Simulations for various cases are made. The analysis and the simulation results indicated that, the best base points are the abstract base points determined via the linear normal mode, which would eliminate the third order terms containing velocity (for cubic systems) or quadratic terms (for quadratic systems) in equations of the NNM oscillators. A better invariance of the NNMs would also be maintained with such base points. The best scheme of normalization is the norm-one scheme that would minimize the numerical errors.     

A CFD assisted control system design with applications to NO x control in a FGR furnace

In this paper, a novel technique to design control systems for industrial processes with non-linear distributed parameters is proposed. The technique utilizes computational fluid dynamics (CFD) simulation to extract the most essential characteristics from the non-linear industrial process, and then represent them as a set of linear dynamic models around a specific operating point. Based on the linear dynamic representation, a closed-loop feedback linear control system can be designed to maintain the desired performance for the system around the chosen operating point. To illustrate such a design process, an industrial reheating furnace with flue gas recirculation (FGR) is selected herein. The method involves the numerical solution of the partial differential equations describing the fluid flow, heat transfer and combustion process in the furnace. The resulting dynamic relations between the furnace inputs and outputs can then be represented in terms of a multi-input and multi-output transfer function matrix. The objective of the control system is then to maintain the optimally selected furnace operating conditions and compensate for any deviations caused by disturbances to minimize the nitric oxides (NO _x ) emission through feedback mechanisms. The performance of the closed-loop controlled furnace is evaluated not only in the linear domain, but also with the detailed full-scale non-linear CFD model. The results have shown that the proposed method is viable and the designed control system can indeed minimize the deviation of the furnace from the desired operating conditions and hence to prevent any excessive NO _x formation in the combustion process.     

Surviving conduction symmetries in non-linear response

In linear response, the electric conductance of mesoscopic, two-terminal devices is symmetric with respect to the direction of an external magnetic field. The conductance symmetry, in general, breaks down in the non-linear regime of transport. Here we consider semiconductor quantum dots and show certain symmetries survive in the non-linear conductance with respect to the bias voltage and magnetic field that can be measured.     

Third-order non-linear susceptibility in a three-level QD system

Third-order non-linear susceptibility is derived for a three-level quantum dot system. Then the total absorption (linear and non-linear) for InGaAsP three-level quantum dot systems is calculated at various parameters (wetting layer composition, pump power, quantum size effect and dephasing linewidth). The spectral hole appears at low power with increasing Ga mole-fraction in the wetting layer.     

具有非线性阻尼涨落的线性谐振子的随机共振

较之于线性噪声, 非线性噪声更广泛地存在于实际系统中, 但其研究远不能满足实际情况的需要. 针对作为非线性阻尼涨落噪声基本构成成分的二次阻尼涨落噪声, 本文考虑了周期信号与之共同作用下的线性谐振子, 关注这类具有基本意义的阻尼涨落噪声的非线性对系统共振行为的影响. 利用Shapiro-Loginov公式和Laplace变换推导了系统稳态响应振幅的解析表达式, 并分析了稳态响应振幅的共振行为, 且以数值仿真验证了理论分析的有效性. 研究发现: 系统稳态响应振幅关于非线性阻尼涨落噪声系数具有非单调依赖关系, 特别是非线性阻尼涨落噪声比线性阻尼涨落噪声更有助于增强系统对外部周期信号的响应程度; 而且, 非线性阻尼涨落噪声比线性阻尼涨落噪声使得稳态响应振幅关于噪声强度具有更为丰富的共振行为; 同时, 二次阻尼涨落噪声使得稳态响应振幅关于系统频率出现真正的共振现象; 而在这些现象和性质中, 非线性噪声项的非线性性质对共振行为起着关键的作用. 显然, 以二次阻尼涨落作为基本形式引入的非线性阻尼涨落噪声, 可以有助于提高微弱周期信号检测的灵敏度和实现对周期信号的频率估计.     

Component mode analysis of a simple non-linear,non-conservative system

A component mode analysis is carried out based upon the use of constraint conditions and Lagrange multipliers to treat exemplary physical systems with non-linear damping. Both viscous and dry friction dampers attached to a linear elastic (beam) system are examined. The method is shown to possess conceptual and computational advantages by its ability to reduce the analysis of a multimode system to a small number of degrees of freedom equal to that associated with the non-linear component.